Theorem pmtrsn 19223

Description: The value of the transposition generator function for a singleton is empty, i.e. there is no transposition for a singleton. This also holds for 𝐴 ∉ V, i.e. for the empty set {𝐴} = ∅ resulting in (pmTrsp‘∅) = ∅. (Contributed by AV, 6-Aug-2019.)

Assertion

Ref	Expression
pmtrsn	⊢ (pmTrsp‘{𝐴}) = ∅

Proof of Theorem pmtrsn

Dummy variables 𝑝 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 5376	. . 3 ⊢ {𝐴} ∈ V
2		eqid 2736	. . . 4 ⊢ (pmTrsp‘{𝐴}) = (pmTrsp‘{𝐴})
3	2	pmtrfval 19154	. . 3 ⊢ ({𝐴} ∈ V → (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
4	1, 3	ax-mp 5	. 2 ⊢ (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
5		eqid 2736	. . . . 5 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
6	5	dmmpt 6178	. . . 4 ⊢ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
7		2on0 8383	. . . . . . . . 9 ⊢ 2o ≠ ∅
8		ensymb 8863	. . . . . . . . . 10 ⊢ (∅ ≈ 2o ↔ 2o ≈ ∅)
9		en0 8878	. . . . . . . . . 10 ⊢ (2o ≈ ∅ ↔ 2o = ∅)
10	8, 9	bitri 274	. . . . . . . . 9 ⊢ (∅ ≈ 2o ↔ 2o = ∅)
11	7, 10	nemtbir 3037	. . . . . . . 8 ⊢ ¬ ∅ ≈ 2o
12		snnen2o 9102	. . . . . . . 8 ⊢ ¬ {𝐴} ≈ 2o
13		0ex 5251	. . . . . . . . 9 ⊢ ∅ ∈ V
14		breq1 5095	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑦 ≈ 2o ↔ ∅ ≈ 2o))
15	14	notbid 317	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (¬ 𝑦 ≈ 2o ↔ ¬ ∅ ≈ 2o))
16		breq1 5095	. . . . . . . . . 10 ⊢ (𝑦 = {𝐴} → (𝑦 ≈ 2o ↔ {𝐴} ≈ 2o))
17	16	notbid 317	. . . . . . . . 9 ⊢ (𝑦 = {𝐴} → (¬ 𝑦 ≈ 2o ↔ ¬ {𝐴} ≈ 2o))
18	13, 1, 15, 17	ralpr 4648	. . . . . . . 8 ⊢ (∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o ↔ (¬ ∅ ≈ 2o ∧ ¬ {𝐴} ≈ 2o))
19	11, 12, 18	mpbir2an 708	. . . . . . 7 ⊢ ∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o
20		pwsn 4844	. . . . . . . 8 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}
21	20	raleqi 3307	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o ↔ ∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o)
22	19, 21	mpbir 230	. . . . . 6 ⊢ ∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o
23		rabeq0 4331	. . . . . 6 ⊢ ({𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} = ∅ ↔ ∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o)
24	22, 23	mpbir 230	. . . . 5 ⊢ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} = ∅
25	24	rabeqi 3416	. . . 4 ⊢ {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
26		rab0 4329	. . . 4 ⊢ {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = ∅
27	6, 25, 26	3eqtri 2768	. . 3 ⊢ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅
28		mptrel 5767	. . . 4 ⊢ Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
29		reldm0 5869	. . . 4 ⊢ (Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅))
30	28, 29	ax-mp 5	. . 3 ⊢ ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅)
31	27, 30	mpbir 230	. 2 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅
32	4, 31	eqtri 2764	1 ⊢ (pmTrsp‘{𝐴}) = ∅

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1540   ∈ wcel 2105  ∀wral 3061  {crab 3403  Vcvv 3441   ∖ cdif 3895  ∅c0 4269  ifcif 4473  𝒫 cpw 4547  {csn 4573  {cpr 4575  ∪ cuni 4852   class class class wbr 5092   ↦ cmpt 5175  dom cdm 5620  Rel wrel 5625  ‘cfv 6479  2_oc2o 8361   ≈ cen 8801  pmTrspcpmtr 19145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-1o 8367  df-2o 8368  df-er 8569  df-en 8805  df-pmtr 19146

This theorem is referenced by:  psgnsn  19224